Prime number
A natural number (i.e. 1, 2, 3, 4, 5, 6, etc.) is called a prime number (or a prime) if it has exactly two positive divisors, 1 and the number itself. Natural numbers greater than 1 that are not prime are called composite. The first 1X5 prime numbers (all the prime numbers less than 1000) are: :2, 3, 5, 7, E, 11, 15, 17, 1E, 25, 27, 31, 35, 37, 3E, 45, 4E, 51, 57, 5E, 61, 67, 6E, 75, 81, 85, 87, 8E, 91, 95, X7, XE, E5, E7, 105, 107, 111, 117, 11E, 125, 12E, 131, 13E, 141, 145, 147, 157, 167, 16E, 171, 175, 17E, 181, 18E, 195, 19E, 1X5, 1X7, 1E1, 1E5, 1E7, 205, 217, 21E, 221, 225, 237, 241, 24E, 251, 255, 25E, 267, 271, 277, 27E, 285, 291, 295, 2X1, 2XE, 2E1, 2EE, 301, 307, 30E, 315, 321, 325, 327, 32E, 33E, 347, 34E, 357, 35E, 365, 375, 377, 391, 397, 3X5, 3XE, 3E5, 3E7, 401, 40E, 415, 41E, 421, 427, 431, 435, 437, 447, 455, 457, 45E, 465, 46E, 471, 481, 485, 48E, 497, 4X5, 4E1, 4EE, 507, 511, 517, 51E, 527, 531, 535, 541, 545, 557, 565, 575, 577, 585, 587, 58E, 591, 59E, 5E1, 5E5, 5E7, 5EE, 611, 615, 617, 61E, 637, 63E, 647, 655, 661, 665, 66E, 675, 687, 68E, 695, 69E, 6X7, 6E1, 701, 705, 70E, 711, 71E, 721, 727, 735, 737, 745, 747, 751, 767, 76E, 771, 775, 77E, 785, 791, 797, 7X1, 7EE, 801, 80E, 817, 825, 82E, 835, 841, 851, 855, 85E, 865, 867, 871, 881, 88E, 8X5, 8X7, 8XE, 8E5, 8E7, 901, 905, 907, 90E, 91E, 921, 927, 955, 95E, 965, 971, 987, 995, 9X7, 9XE, 9E1, 9E5, 9EE, X07, X0E, X11, X17, X27, X35, X37, X3E, X41, X45, X4E, X5E, X6E, X77, X87, X91, X95, X9E, XX7, XXE, XE7, XEE, E11, E15, E1E, E21, E25, E2E, E31, E37, E45, E61, E67, E6E, E71, E91, E95, E97, EX5, EE5, EE7 Except 2 and 3, all primes end with 1, 5, 7 or E. The first k'' such that all of 10''k, 10''k'' + 1, 10''k'' + 2, ..., 10''k'' + E are composite is 38, i.e. all of 380, 381, 382, ..., 38E are composite. (numbers k'' such that all of 10''k, 10''k'' + 1, 10''k'' + 2, ..., 10''k'' + E are composite are 38, 5X, 60, 62, 89, 93, 94, E0, E5, E8, ...) The maximal gaps of primes less than 1000 is (927, 955), the gap (the difference of these two numbers) is 2X (i.e. there are 29 consecutive composites between 927 and 955). The density of primes end with 1 is a relatively low (< 1/4), but the density of primes end with 5, 7 and E are nearly equal (all are a little more than 1/4). (i.e. for a given natural number N'', the number of primes end with 1 less than ''N is usually smaller than the number of primes end with 5 (or 7, or E) less than N'') e.g. For all 1426 primes < 10000, there are 3E8 primes (2E.3%) end with 1, 410 primes (30.3%) end with 5, 412 primes (30.5%) end with 7, 406 primes (2E.E%) end with E. It is conjectured that for every natural number ''N ≥ 10, the number of primes end with 1 less than N'' is smaller than the number of primes end with 5 (or 7, or E) less than ''N. (Note: the percentage in this section are also in duodecimal, i.e. 20% means 0.2 or 20/100 = 1/6, 36% means 0.36 or 36/100 = 7/20, 58.7% means 0.587 or 587/1000) 13665 is the smallest prime p'' such that the number of primes end with 1 or 5 ≤ ''p is more than the number of primes end with 3, 7 or E ≤ p'' (see , of course, 3 is the only prime ends with 3). Besides, 9X03693X831 is the smallest prime ''p such that the number of primes end with 1 or 7 ≤ p'' is more than the number of primes end with 2, 5 or E ≤ ''p (see , of course, 2 is the only prime ends with 2). Question: What is the smallest prime p'' such that the number of primes end with 1 ≤ ''p is more than the number of primes end with d'' ≤ ''p for at least one of d = 5, 7 or E? All squares of primes (except 2 and 3) end with 1. There are 2X primes between 0 and 100, 23 primes between 100 and 200, 1X primes between 200 and 300, 1X primes between 300 and 400, 1E primes between 400 and 500, 1X primes between 500 and 600, 16 primes between 600 and 700, 1X primes between 700 and 800, 18 primes between 800 and 900, 16 primes between 900 and X00, 1X primes between X00 and E00, 17 primes between E00 and 1000. There are about N''/ln(''N) primes less than N'', where ln is the natural logarithm, i.e. the logarithm with base ''e = 2.875236069821... (see prime number theorem), thus there are about \frac{10^n}{n \cdot ln(10)} primes less than 10''n'' (i.e. with at most n'' digits), and ln(10) = 2.599E035E8169... In the following table, numbers shaded in cyan are primes. The twin prime pairs below 100 are (3, 5), (5, 7), (E, 11), (15, 17), (25, 27), (35, 37), (4E, 51), (5E, 61), (85, 87), (8E, 91), (E5, E7), and there are only 47 twin prime pairs below 1000. Except (3, 5), all twin prime pairs end with (5, 7) or (E, 1), and it is conjectured that the density of them are the same. Largest known prime number The largest known prime is a Mersenne prime 223,7XE,125−1, it has 7,870,133 digits, the value of this prime is 174,164,383,313,720,093,6E1,098,...,997,379,395,52E,129,858,481,227 (only show the first 20 digits and the last 20 digits) Classes of prime numbers * By formula: ** Fermat primes: of the form 22''n + 1 *** Generalized Fermat primes base b'': of the form ''b''2''n + 1 ** Mersenne primes: of the form 2''p'' − 1 with prime p'' *** Generalized Mersenne primes (Generalized repunit primes) base ''b: of the form (b''p'' − 1)/(b'' − 1) ** Double Mersenne primes: of the form 22''p − 1 − 1 with prime p'' ** Wagstaff primes: of the form (2''p + 1)/3 with odd prime p'' *** Generalized Wagstaff primes base ''b: of the form (b''p'' + 1)/(b'' + 1) with odd prime ''p ** Factorial primes: of the form n''! ± 1 ** Primorial primes: of the form ''p# ± 1 with prime p'' ** Proth primes: of the form ''k×2''n'' + 1 with odd k'' and 2''n > k'' *** Generalized Proth primes base ''b: of the form k''×''bn'' + 1 with ''k not divisible by b'' and ''bn'' > ''k ** Riesel primes: of the form k''×2''n − 1 with odd k'' and 2''n > k'' *** Generalized Riesel primes base ''b: of the form k''×''bn'' − 1 with ''k not divisible by b'' and ''bn'' > ''k ** Pythagorean primes: of the form 4''n'' + 1 (i.e. end with 1 or 5) ** Pierpont primes: of the form 2''m''×3''n'' + 1 ** Cullen primes: of the form n''×2''n + 1 *** Generalized Cullen primes base b'': of the form ''n×''b'n'' + 1 with n'' ≥ ''b−1 ** Woodall primes: of the form n''×2''n − 1 *** Generalized Woodall primes base b'': of the form ''n×''b'n'' − 1 with n'' ≥ ''b−1 * By size: ** Titanic primes: at least 1,000 (a thousand) digits ** Gigantic primes: at least 10,000 (a myriad) digits ** Megaprimes: at least 106 (a million) digits ** Bevaprimes: at least 109 (a billion) digits ** Teraprimes: at least 1010 (a trillion) digits ** Petaprimes: at least 1013 (a quadrillion) digits ** Exaprimes: at least 1016 (a quintillion) digits ** Zettaprimes: at least 1019 (a sextillion) digits ** Yottaprimes: at least 1020 (a septillion) digits There are only 26 known megaprimes and 2 known mega-PRPs (no bevaprimes and above are known), the smallest known megaprime is 3×23,762,E96+1, it has 1,018,16E digits, it is a Proth prime and a Pierpont prime, besides, both of the two known mega-PRPs are Wagstaff PRPs (of the form (2''n''+1)/3), the exponents n of them are 4,578,17E and 4,58X,897. 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